1. **State the problem:** We have a parametric curve defined by the equations: $$x = 9 + 2t^2$$ $$y = 4t$$ with the parameter $t$ ranging from $0$ to $2$. 2. **Goal:** Understand the shape and position of the curve by analyzing the parametric equations. 3. **Express $y$ in terms of $t$:** $$y = 4t$$ 4. **Express $x$ in terms of $t$:** $$x = 9 + 2t^2$$ 5. **Eliminate parameter $t$ to find $y$ as a function of $x$:** From $y = 4t$, solve for $t$: $$t = \frac{y}{4}$$ Substitute into $x$: $$x = 9 + 2\left(\frac{y}{4}\right)^2 = 9 + 2\frac{y^2}{16} = 9 + \frac{y^2}{8}$$ 6. **Rewrite the equation:** $$x - 9 = \frac{y^2}{8} \implies y^2 = 8(x - 9)$$ 7. **Interpretation:** This is a parabola opening to the right with vertex at $(9,0)$. 8. **Parameter range:** Since $t$ ranges from $0$ to $2$, $y$ ranges from $0$ to $8$ and $x$ ranges from $9$ to $9 + 2(2)^2 = 9 + 8 = 17$. 9. **Position hint (top-left):** The curve starts at $(9,0)$ when $t=0$ and moves to $(17,8)$ when $t=2$, moving upward and rightward. **Final answer:** The parametric curve is a right-opening parabola segment from $(9,0)$ to $(17,8)$ described by $$y^2 = 8(x - 9), \quad 0 \leq y \leq 8.$$